Question

The coordinates of the vertices of quadrilateral DEFG are D(−2, 5) , E(2, 4) , F(0, 0) , and G(−4, 1) .

Answer 1

Answer:

a.Quadrilateral DEFG is not a rhombus because opposite sides are parallel but the four sides do not all have the same length.

Explanation:

To determine if the opposite sides are parallel, we find the slope of the line that passes through each pair of points.  The formula for slope is:

$m=\frac{y_2-y_1}{x_2-x_1}$

The slope of DE is:

m = (5-4)/(-2-2) = 1/-4 = -1/4

The slope of EF is:

m = (4-0)/(2-0) = 4/2 = 2

The slope of FG is:

m = (0-1)/(0--4) = -1/4

The slope of GD is:

m = (1-5)/(-4--2) = -4/-2 = 2

The slopes of opposite sides are parallel.

Now we must determine if the sides are congruent.  To do this, we use the distance formula:

$d = \sqrt{(y_2-y_1)^2+(x_2-x_1)^2}$

The length of DE is:

$d=\sqrt{(5-4)^2+(-2-2)^2}=\sqrt{1^2+(-4)^2}=\sqrt{1+16}=\sqrt{17}$

The length of EF is:

$d=\sqrt{(4-0)^2+(2-0)^2}=\sqrt{4^2+2^2}=\sqrt{16+4}=\sqrt{20}$

The length of FG is:

$d=\sqrt{(0-1)^2+(0--4)^2}=\sqrt{(-1)^2+4^2}=\sqrt{1+16}=\sqrt{17}$

The length of GD is:

$d=\sqrt{(1-5)^2+(-4--2)^2}=\sqrt{(-4)^2+(-2)^2}=\sqrt{16+4}=\sqrt{20}$

Not all sides are congruent, so this is not a rhombus.

Answer 2

Answer: The correct option is a.Quadrilateral DEFG is a rhombus because opposite sides are parallel and all four sides have the same length.

Explanation:

It is given that  coordinates of the vertices of quadrilateral DEFG are D(−2, 5) , E(2, 4) , F(0, 0) , and G(−4, 1) .

The distance formula for two points is given below,

$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

$DE=\sqrt{(2+2)^2+(4-5)^2}=\sqrt{17}$

$EF=\sqrt{(0-2)^2+(0-4)^2}=\sqrt{20}$

$FG=\sqrt{(-4-0)^2+(1-0)^2}=\sqrt{17}$

$DG=\sqrt{(-4+2)^2+(1-5)^2}=\sqrt{20}$

Slope formula,

$m=\frac{y_2-y_1}{x_2-x_1}$

Slope of DE,

$m_1=\frac{4-5}{2+2}= \frac{-1}{4}$

Slope of EF,

$m_2=\frac{0-4}{0-2}= 2$

Slope of FG,

$m_3=\frac{1-0}{-4-0}= \frac{-1}{4}$

Slope of DG,

$m_4=\frac{1-5}{-4+2}= 2$

The slope of opposite sides are equal, it means the opposite sides are parallel.

All four sides do not have the same length. It means it is not a rhombus.

From the figure it is noticed that the opposite sides are parallel.

Therefore, the correct option is a.Quadrilateral DEFG is a rhombus because opposite sides are parallel and all four sides have the same length.